Transformasi Geometri Bagian 1 - Translasi (Pergeseran) Matematika Wajib Kelas 11

Introduction

In this tutorial, we will explore the concept of geometric transformations, specifically focusing on translation, which is a fundamental topic in mathematics for class XI. Understanding translation will help you manipulate geometric shapes in a Cartesian plane, laying the groundwork for more complex transformations such as rotation and dilation.

Step 1: Understanding Translation

Translation, or pergeseran, refers to the movement of points, lines, or curves in a geometric space without altering their shape or size. This transformation can be described with the following key points:

• Definition: A translation moves every point of a shape the same distance in a specified direction.
• Notation: If a point P has coordinates (x, y), after translation by a vector (a, b), the new coordinates P' will be (x + a, y + b).

Practical Tip

Visualizing translations on graph paper or using graphing software can help solidify your understanding of how shapes move.

Step 2: Translating Points

To perform a translation of a point using a vector, follow these steps:

1. Identify the original point coordinates, (x, y).
2. Determine the translation vector (a, b).
3. Apply the translation formula:
   • New coordinates: P' = (x + a, y + b).

Example

For point P(3, 4) and a translation vector (2, 3):

• New coordinates: P' = (3 + 2, 4 + 3) = (5, 7).

Step 3: Translating Lines and Curves

Translating lines and curves involves shifting all points along these shapes according to the same vector.

1. Select the line or curve represented by its equation or set of points.
2. Translate each point using the vector (a, b) as described in Step 2.
3. If dealing with a linear equation, adjust the equation accordingly to reflect the new position.

Common Pitfall

Ensure that all points are translated uniformly; otherwise, the shape may distort or misrepresent the original.

Step 4: Composing Translations

Multiple translations can be combined into a single transformation. This is called composition.

1. Perform the first translation on the original shape.
2. Take the resulting shape and apply the second translation.
3. The cumulative effect can be expressed as a single vector:
   • If T1 = (a1, b1) and T2 = (a2, b2), then the combined translation T = (a1 + a2, b1 + b2).

Real-World Application

Understanding translation is essential in computer graphics, robotics, and animation, where objects need to move smoothly from one position to another.

Conclusion

In this tutorial, we covered the basics of geometric translation, including how to translate points, lines, and curves, as well as how to compose multiple translations. Mastering these concepts will enhance your understanding of more complex transformations in geometry. As a next step, practice translating different shapes and explore how these transformations affect their positions on the Cartesian plane.